For finishing worn steel edges of a ski it is known (DE 39 14 977 A1) to use a pot-shaped grinding wheel driven by a motor with an axis of rotation inclined under an acute angle with respect to the feed direction and to the running surface of the ski, so that the ring-shaped end face of the grinding wheel rests against the outer machining surface of the steel edge merely in a peripheral portion. With such a setting of the grinding wheel there can be achieved a feed following the course of the side face of the steel edges waisted in longitudinal direction, but the grinding results remain unsatisfactory, because due to the waisted course of the steel edge the set angle changes over the length of the steel edge. Moreover, with such an apparatus merely the side faces of the steel edges, but not their machining surfaces on the side of the running surface can be ground.